2010/05/27 by Ido Ben‐Eliezer, Michael Krivelevich, Ben-Eliezer, Ido +3 · 1 citation
Computer Science · Mathematics · #Advanced Graph Theory Research #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.1005.5171
openalex publication_date 2010/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a graph H, the size Ramsey number re(H,q) is the minimal number m for which there is a graph G with m edges such that every q-coloring of G contains a monochromatic copy of H. We study the size Ramsey number of the directed path of length n in oriented graphs, where no antiparallel edges are allowed. We give nearly tight bounds for every fixed number of colors, showing that for every q≥ 1 there are constants c1 = c1(q),c2 such that \fracc1(q) n2q(log n)1/q(loglog n)(q+2)/q ≤ re(\overrightarrowPn,q+1) ≤ c2 n2q(log n)2. Our results show that the path size Ramsey number in oriented graphs is asymptotically larger than the path size Ramsey number in general directed graphs. Moreover, the size Ramsey number of a directed path is polynomially dependent in the number of colors, as opposed to the undirected case. Our approach also gives tight bounds on re(\overrightarrowPn,q) for general directed graphs with q ≥ 3, extending previous results.